How to move an element symmetrically in edit mode
Perhaps the simplest and most surefire way is create a mirror modifier.
Split your object down the middle in Edit Mode by pressing Ctrl + r to generate a new edge loop going all the way around the object at its center. After activating this shortcut, float your cursor a bit along a horizontal edge near the top or bottom center of your object until you see a vertical division appear which snaps to the centerline of the object. Then left-click immediately afterwards one single time to commit to that division, then right-click one single time to cancel any further mode actions, which will then commit to this edit.
Now remove half of your model. Box select the half to be removed. Do this in an orthographic, wireframe front view, to make sure you don't miss any of the desired parts. Also do it with Face Selection mode active (so that you are sure to select faces and not edges or vertices). Then delete these faces so that only half of your model remains.
Finally, create a mirror modifier from the modifier tab (wrench icon) in the Properties Panel. Assuming your object's pivot center is properly aligned with the cut you made (which it typically will be by default), the mirrored half will be positioned correctly in relation to the original half, without the need for further settings changes. The mirrored half will perfectly mimic the other half whenever you tweak it in Edit mode.
When you are done modeling, you should probably apply the mirror modifier, which will make the mirrored results permanent, and give you two proper sides to the object afterwards. Note that you cannot do this apply function while still in Edit mode.